Quasistatic Modeling of Concentric Tube Robots with External Loads.
ABSTRACT Concentric tube robots are a subset of continuum robots constructed by combining precurved elastic tubes. As the tubes are rotated and translated with respect to each other, their curvatures interact elastically, enabling control of the robot's tip configuration as well as the curvature along its length. This technology is projected to be useful in many types of minimally invasive medical procedures. Because these robots are flexible by design, they deflect considerably when applying forces to the external environment. Thus, in contrast to rigidlink robots, their kinematic and static force models are coupled. This paper derives a multitube quasistatic model that relates tube rotations and translations together with externally applied loads to robot shape and tip configuration. The model can be applied in robot design, procedure planning as well as control. For validation, the multitube model is compared experimentally to a computationallyefficient singletube approximate model.

Conference Paper: Robot strings: Long, thin continuum robots
[Show abstract] [Hide abstract]
ABSTRACT: We describe and discuss the development of long, thin, continuous “stringlike” robots aimed at Space exploration missions. These continuous backbone “continuum” robots are inspired by numerous biological structures, particularly vines, worms, and the tongues of animals such as the anteater. The key novelty is the high lengthtodiameter ratio of the robots. This morphology offers penetration into, and exploration of, significantly narrower and deeper environments than accessible using current robot technology. In this paper, we introduce new design alternatives for long thin continuum robots, based on an analysis and extension of three core existing continuum robot design types. The designs are evaluated based on their mechanical feasibility, structural properties, kinematic simplicity, and degrees of freedom.Aerospace Conference, 2013 IEEE; 01/2013  SourceAvailable from: Hongliang Ren
Conference Paper: Statics Modeling of an Underactuated WireDriven Flexible Robot Arm
BioRob 2014, IEEE/RASEMBS International Conference on Biomedical Robotics and Biomechatronics; 01/2014 
Conference Paper: Position control of concentrictube continuum robots using a modified Jacobianbased approach
[Show abstract] [Hide abstract]
ABSTRACT: Concentrictube robots can offer dexterous positioning even in a small constrained environment. This technology turns out to be beneficial in many classes of minimally invasive procedures. However, one of the barriers to the practical use of a concentrictube robot is the design of a realtime control scheme. In previous work by the authors, a computationally efficient torsionally compliant kinematic model of a concentrictube robot was developed. Using this computationally fast technique and deriving the robot's Jacobian, a new position control approach is proposed in this paper. This mechanism provides computational efficiency as well as good tracking accuracy. To evaluate the performance, experiments were conducted, and the results obtained demonstrate the feasibility of enabling the robot's tip to perform trajectory tracking in real time.Robotics and Automation (ICRA), 2013 IEEE International Conference on; 01/2013
Page 1
The 2010 IEEE/RSJ International Conference on
Intelligent Robots and Systems
October 1822, 2010, Taipei, Taiwan
Abstract—Concentric tube robots are a subset of continuum
robots constructed by combining precurved elastic tubes. As
the tubes are rotated and translated with respect to each other,
their curvatures interact elastically, enabling control of the
robot’s tip configuration as well as the curvature along its
length. This technology is projected to be useful in many types
of minimally invasive medical procedures. Because these robots
are flexible by design, they deflect considerably when applying
forces to the external environment. Thus, in contrast to rigid
link robots, their kinematic and static force models are
coupled. This paper derives a multitube quasistatic model that
relates tube rotations and translations together with externally
applied loads to robot shape and tip configuration. The model
can be applied in robot design, procedure planning as well as
control. For validation, the multitube model is compared
experimentally to a computationallyefficient singletube
approximate model.
I. INTRODUCTION
HE goal of minimally invasive surgery (MIS) is to
interact with tissue deep inside the body while
minimizing collateral damage to surrounding tissues. In
contrast to open surgery in which access is gained by
making large incisions, MIS involves entering the body
through small incisions and, whenever possible, following
natural passages through the tissues to reach the surgical
site. Manual and robotic catheters are successful examples
of an MIS instrument technology which have been
specifically developed for procedures inside the vasculature
[1],[2].
There are many medical procedures that could benefit
from an instrument technology with the ability of catheters
to follow complex curves, but which require much more tip
stiffness than that of a catheter. These include structural
repairs inside the heart and tissue removal inside the brain.
Few instrument technologies exist, however, that possess
significant tip stiffness in combination with the ability to
assume 3D curves inside the body. Conventional surgical
Manuscript received February 28, 2010. This work was supported by the
National Institutes of Health under grants R01HL073647 and
R01HL087797.
J. Lock is with Biomedical Engineering, Boston University, Boston, MA
02215 USA (lockj@bu.edu).
G. Laing is with Mechanical Engineering, Boston University, Boston,
MA 02215 USA (glaing@bu.edu).
M. Mahvash is with Harvard Medical School, Boston MA 02115 USA
(mohsenmahvash@gmail.com).
P. Dupont is with Cardiovascular Surgery, Children’s Hospital Boston,
Harvard Medical School, Boston MA 02115 USA
(Pierre.Dupont@childrens.harvard.edu).
robots, for example, possess high stiffness, but consist of
straight shafts comparable to traditional laparoscopic tools
[3]. To address this shortcoming, bending snakelike robotic
extensions have been proposed and constructed for
mounting at the tip of the straight shaft [4]. A novel,
alternate approach consists of a robotic sheath that can be
extended along a 3D curve [5].
Concentric tube robots offer a good compromise between
shape control and stiffness. As illustrated by the example of
Fig. 1, they can be constructed to possess a full six degrees
of freedom at their tip while also enabling control of
curvature along their length. Furthermore, they can be
constructed with diameters comparable to catheters and
lengths sufficient to reach anywhere inside the body while
achieving a tip stiffness several orders of magnitude greater
than that of a catheter. The lumen of the innermost tube can
house additional tubes and wires for controlling articulated
tipmounted tools.
Fig. 1. Concentric tube robot comprised of four telescoping sections that
can be rotated and translated with respect to each other.
Concentric tube robots, like steerable catheters [1],[2] and
snakelike multibackbone devices [4], are continuum
robots. In comparison to traditional robot arms, this class of
robots lacks distinct links and joints. Continuum robots
possess the shape of a smooth curve whose curvature can be
controlled by adjusting the internal deformation of
mechanically coupled elastic components of the body.
Consequently, the kinematic modeling of continuum
robots cannot be formulated solely in terms of constrained
motion between rigid bodies, but must also incorporate
deformation modeling of the elastic components [1],[4],[6]
[9]. For concentric tube robots, the deformation is that of the
individual tubes [6][9].
Owing to both the complexity of the modeling problem as
well as to the desire to derive numerically efficient models
for realtime control, a succession of kinematic models of
increasing complexity have been proposed over the last few
years as described in [8][9]. While providing significantly
improved accuracy over earlier models, these new models
are considerably more complex. They consist of second
Quasistatic Modeling of Concentric Tube Robots
with External Loads
Jesse Lock, Genevieve Laing, Mohsen Mahvash and Pierre E. Dupont, Senior Member, IEEE
T
9781424466764/10/$25.00 ©2010 IEEE2325
Page 2
order differential equations with split boundary conditions.
To achieve computational efficiency, these equations can be
precomputed over the workspace and stored either in the
form of a functional approximation or as a lookup table [8].
The inverse kinematic problem can be solved efficiently by
root finding on the approximate forward solution [8].
Alternately, an inverse functional approximation or lookup
table can be similarly constructed.
While the kinematic models of [8][9] assume that there
is no external loading applied to the robot (see [10] for an
exception), applications in minimally invasive surgery can
be expected to involve loads applied along the robot’s length
as well as at its tip. Unlike robots whose links can be
approximated as rigid bodies, however, the kinematic and
static force models of continuum robots cannot be
decoupled.
Thus, when considering the important case of external
loads applied to the robot, the model for implementing
position, force or impedance control takes the form of a
coupled 3D beambending problem in which the kinematic
input variables (tube rotations and displacements at the
proximal end) enter the problem as a subset of the boundary
conditions. The remaining boundary conditions are
comprised of point forces and torques applied to the distal
ends of the tubes. Contact along the robot’s length (e.g.,
with tissue) generates additional distributed and point loads.
In contrast to the models of [8][9], the inclusion of
external loading significantly increases the number of state
variables that must be integrated along the lengths of the
tubes. As an alternative to this fullorder model, a
computationallyefficient approximate model that can be
applied to all types of continuum robots has been proposed
and successfully implemented for concentrictube robot
stiffness control [11][13]. In this approach, the continuum
robot is modeled as a single Cosserat rod with properties
along its length corresponding to the composite stiffnesses
and initial curvatures of the unloaded robot.
The contributions of this paper are the derivation of a
multitube quasistatic model as well as a computational and
experimental comparison of the multitube model with the
singletube model of [11][13]. The paper is arranged as
follows. Section II derives the multitube externallyloaded
model. Section III presents the simplified singletube
approximate model. Section IV provides an experimental
comparison of the models. Conclusions are presented in
Section V.
II. QUASISTATIC MULTITUBE MODEL
The multitube model derived here can be interpreted as
an extension of the unloaded model presented in [8]. It
includes bending and torsion for an arbitrary number of
tubes whose curvature and stiffness can vary with arc length.
Effects that are neglected include shear of the cross section,
axial elongation, nonlinear constitutive behavior and friction
between the tubes. Note that these effects are neglected, but
are not necessarily all negligible.
In the remainder of the paper, subscript indices
1,2,...,
in are used to refer to individual tubes with tube 1
being outermost and tube n being innermost. Arc length, s,
is measured such that s = 0 at the proximal end of the tubes.
The total length of each tube is designated by Li.
As illustrated in Fig. 2, for two tubes, material coordinate
frames for each cross section can be defined as a function of
arc length s along tube i by defining a single frame at the
proximal end,
(0)
, such that its z axis is tangent to the
tube’s centerline. Under the unrestrictive assumption that the
tubes do not possess initial material torsion, the frame,
is obtained by sliding
(0)
without rotation about its z axis (i.e., a Bishop frame [14]).
As the tubes move, bend and twist, these material frames act
as body frames tracking the displacements of their cross
sections. It is also useful to define a reference frame,
which displaces with the cross sections, but does not rotate
about its z axis under tube torsion.
As the
centerline, it experiences a bodyframe angular rate of
change per unit arc length given by
é
ëê
in which (,)
ix iy
uu
are the components of curvature due to
bending and
iz
u is the curvature component due to torsion. A
circumflex on a curvature component is used to designate
the initial precurvature of a tube.
The kinematic input variables consist of the rotation and
translation of each tube about and along the common
centerline of the combined tubes. The rotation angle, qi(s) ,
is defined as the z axis rotation angle from frame
frame
( )
iF
( )
iF s ,
iF
along the tube centerline
0( )F s ,
thi tube’s coordinate frame
( )
iF s slides down its
ui(s)=
uix(s)uiy(s)uiz(s)
ù
ûú
T (1)
0( )F s to
iF s . The translation variable, li, is defined as the
arc length distance from frame F0(0) to the initially
coincident frame Fi(0) . In the rest of the paper, all vector
quantities associated with tube i , e.g.,
with respect to frame
( )
robot, e.g., net bending moment, are written with respect to
frame
F s .
As shown in the figure, insertion of one tube inside the
other causes each to bend and twist along their length. The
application of externally applied wrenches generates
additional bending and twisting of the tubes.
( )
iu s , are written
iF s . Vectors associated with the
0( )
A. Derivation of Multitube Model
The quasistatic model including external loading can be
derived by combining three equations – a constitutive model
relating bending moments to changes in curvature of
individual tubes, the equilibrium of bending moments and
shear forces on the cross section of the assembled tubes, and
a compatibility equation relating the individual curvatures of
the assembled tubes. Additional equations are needed to
compute the net shear force and bending moment on the
robot as a function of arc length.
2326
Page 3
The constitutive model and compatibility equations are
independent of the external loading and so are identical to
those of the unloaded kinematic model presented in [8]. The
equilibrium equation of [8], however, must be modified to
include the net bending moment arising from external loads.
Furthermore, to compute net bending moment, new
differential equations must be introduced to compute both it
and net shear force. Each is described below.
Fig. 2. Tube coordinate frames are denoted
( )
iF s . The relative zaxis twist
( ) ( )
.
angle between tube frame
(1) Constitutive Model: When a tube with initial
curvature ˆ ( )
bending moment is generated. Assuming linear elastic
behavior, the bending moment vector
along tube i is given by
( )
ii
m s K u s
Given the coordinate frame convention described above, all
vectors are expressed with respect to frame
the frameinvariant stiffness tensor given by
00
00
00
iz
k
in which
moment of inertia,
is the shear modulus of tube i.
(2) Compatibility of Deformations: Assuming that the
clearance between each pair of adjacent tubes is just
sufficient to enable relative motion, all tubes must conform
to the same final xy (bending) curvature. Each tube is free,
however, to twist independently about its z axis. The z
component of curvature,
( )
i
u s
change of twist angle with respect to arc length,
0( )F s and frame
iF s is
is
iu s is deformed to a different curvature
( )
iu s , a
( )
i
m s at any point s
ˆ
( ) ( )
ii
u s
(2)
( )
iF s , and
i
K is
00
00
00
ixi i
i iyi i
ii
k E I
KkE I
J G
(3)
iE is the modulus of elasticity,
iJ is the polar moment of inertia and
iI is the area
i
G
z
, equates to the rate of
( )
is
,
( )s( )
ii
z
u s
. (4)
The resulting bending curvatures can be equated when
written in the same frame. Expressing tube curvatures in
terms of the robot frame curvature,
in which
( ) (3)
zi
RSO
is a rotation about the z axis by
angle
[0,0,1]T
.
(3) Equilibrium of Bending Moments: On each cross section,
the bending moments in each tube must sum to the robot’s
net bending moment,
m s , generated by the external
loading.
n
m sR
As in (5),
( )
zi
R is used to transform tube bending moments
from frame
( )F s .
Combining (2) and (6) expresses net bending moment in
terms of tube curvatures,
n
m sRs K s u s
Solving (5) and (7) for
u s provides an expression for
robot curvature in terms of initial tube curvatures and net
bending moment,
1
( )( )
i
i
Since frame
F s by definition does not rotate about its
z axis,
0
0
z
u
, and so this equation can written in two
parts as
0u , results in
( )
i
s e
0
( )( )( )
T
ziiz
u sRu s
(5)
i and
ze
0( )
0
1
( )( )
( )
zii
i
m s
(6)
iF s to frame
0( )
0
1
ˆ
( )( ( ))
i
( )( ( )( ))
ziii
i
u s
. (7)
0( )
0
1
0
1
ˆ
( ( ))
i
( ) ( )( )
s K s e
( )( )
n
n
ziiiiz
i
u sK s
Rs K s u sm s
(8)
0( )
1
00
,
11
,
ˆ
( )( )( ( ))
i
( ) ( )( )
nn
izii
x y
ii
x y
u sK s Rs K s u sm s
(9)
0
11
( )( ) ( ) s
( )s u s ( )
nn
iziiz iz
z
ii
m sksk
(10)
Equations (5) and (9) enable the computation of the x and y
curvatures of all tubes using
,
( )
iz
x y
u sR
An expression is also needed to compute the z curvature of
all tubes,
,1,,
izi
uin
. Such an expression can be
obtained from the equilibrium equation of the special
Cosserat rod model [15][17]. Setting time dependent terms
to zero, the bodyframe equilibrium equations for a curved
rod undergoing distributed loading of t Î3torque per unit
length and f Î 3force per unit length can be applied to
each tube
( )( ) [ ( )] [ ( )]
( )( )
ii
n sf s
0
,
( ) ( )
i
u s
T
x y
(11)
( )
( )0[ ( )]
i
u s
iiiii
i
m s
su sv sm s
n s
(12)
2327
Page 4
Derivatives are with respect to arc length along the rod, s,
and
,
ii
m n Î are the bending moment and shear force
vectors acting on the tube’s cross section. Here, and in the
remainder of the paper, the square brackets on the vectors
iu and
0
[ ]
iiz
uu
u
Consistent with the previous notation,
the angular and linear strain rates per unit arc length,
respectively, experienced by the tube's cross section. Thus,
as described previously,
( )
Similarly, the x and y components of
strain components of the cross section while the z
component is
1
iziz
v
strain. Given the assumptions of negligible shear and
longitudinal strain,
( )0
It can be helpful to note that
to bodyframe angular and linear velocities if time is
substituted for arc length. Wrenches applied at either end of
the rod enter the equations as boundary conditions.
Since tube interaction is limited to distributed forces,
( )0
t= in (12) and, for each tube, it reduces to
[ ]
ii
mu m
To eliminate moments from these equations, we can use the
constitutive model for moments (2) and its derivative to
arrive at
ˆ
( )( )( ( )/
iz izz
ususk sk s
k sk su
3
iv denote the skewsymmetric form
0
0
iz iy
u
ix
iyix
uu
u
(13)
3
( ), ( ) u s v s Î are
ii
iu s has the units of curvature.
( )
iv s are the shear
in which
iz
is the longitudinal
0 1
T
iv s
(14)
( )
iu s and
( )
iv s are analogous
is
[ ]v n
iii
(15)
ˆ
u s
( ))(( )( ))
ˆˆ
(( )/( ))( )
s u
( )
s
( )
s u
( )
s
zzz
xzixiyiyix
u s
u
(16)
Equations (4) and (16) are a set of second order differential
equations for the tubes’ twist angles,
integrated using the algebraic equations (9) and (11). These
equations are identical to those describing the unloaded
kinematic model except that (9) now includes the net
bending moment on the tubes,
(4) Net Bending Moment and Shear Force: While (11)
provides the z component of
components that are needed for (9). To compute net bending
moment as a function of arc length, the equilibrium special
Cosserat model (12) can be applied again, but this time to
the collection of tubes.
( )( )[ ( )] [ ( )]
( )( )n sf s
Since net bending moment on the robot’s cross section
evolves together with net shear force,
simultaneously integrated. Here
externally applied distributed torque and force per unit
length of the robot as shown in Fig. 3.
i , that must be
0( )m s [8].
0( )m s , it is the x and y
00000
0000
( )
( )0[ ( )]u s
m s
s u sv sm s
n s
(17)
0( )n s , both must be
and
f s are the
0( ) s
0( )
Fig. 3. External loading on robot consists of distributed forces,
0( )f s , and
distributed moments,
0( ) s
t
, as well as concentrated forces,
0( )n L , and
concentrated moments,
Robot curvature,
applies to all tubes comprising the robot,
Equations (4),(9)(11),(16)(18) form a set of equations in
the state variables
( ),
ij
s
2,,jn
. Observe that
1
( )
algebraically from (10).
The boundary conditions for the state variables are split
between the proximal and distal ends of the robot.
(0) actuator positions
i
LuL
m L
n L
The x and y components of
( )
and (11). While (10) evaluated at s
expression for the weighted sum of
insufficient to solve for the individual values of
can be resolved by assuming that the total external twisting
moment is applied to a single tube, say tube j. The resulting
values for
( )
iz
uL are given by
0( )/
( )
0
Physically, this situation corresponds to tube j extending
slightly beyond the other tubes so that it comprises the tip of
the robot.
There is, in fact, no reason that the tubes must be of the
same length. The equations above apply to any telescoping
arrangement of tubes in which the stiffness and pre
curvature of each tube can be an arbitrary function of arc
length. This includes discontinuities in both stiffness and
precurvature. Consequently, there is no need to subdivide
the domain during integration over a telescoping
arrangement of tubes. Distal to the physical end of each
tube, its stiffness and curvature can be defined as zero.
0( )m L .
0( )u s , is defined by (8) and since (14)
0( )v s
00 1
T
. (18)
00
( ),s m s n s( ),( )
,
1,,in
;
1
( )s
z
su
can be computed
0
0
( )
( )
( )
( )
body frame external tip moment
body frame external tip force
i iz
(19)
iu L can be computed from (9)
L
( ),
iz
uL i
provides an
1,,n
( )
iz
uL . This
, it is
,
ziz
iz
mLki
i
j
j
u L
. (20)
B. Numerical Solution of Multitube Model
When solving the multitube equations given by (4),(9)(11),
(16)(18) together with boundary conditions defined by (19)
and (20), three issues must be considered. First, the
boundary conditions are split between the ends of the tubes.
Second, integration of (
0
( ) v s
)
3
0
,( )(3)u sso
ÎÎ
to obtain
2328
Page 5
the robot coordinate frame,
that it evolves on
in these equations is expressed in the body coordinate frame,
it is often convenient to express external loads with respect
to a different frame. Each of these issues is addressed in the
paragraphs below.
(1) Split Boundary Conditions: The problem of split
boundary conditions is one that has been addressed with the
unloaded kinematic equations. In fact, the unloaded
equations can be recovered by setting the external loading to
zero [8].
00
( ) ( )s f s m L
t==
While such equations can be solved by a variety of standard
means, one approach is to pose the forward kinematics as a
root finding problem in which guesses of
integrate from
0
sL
until the desire values of
obtained.
(2) Integration on SE(3): Integrating the unloaded
kinematics required integrating tube curvatures with respect
to arc length. Analogous to integrating body frame twist
velocity, numerical integration of ui and vimust preserve
the group structure of SE(3). A variety of numerical
integration methods are available for this purpose
[16],[18],[19].
(3) ExternalLoad Coordinate Frame: It is often desirable
to express the loading in different coordinates than the body
frame coordinates of (19) and (20). In this case, however,
the boundary condition is a function of the shape of the
robot. For example, suppose it is desired to produce a tip
wrench that is specified with respect to the base frame of the
robot, F0(0) . Then the bodyframe tip wrench, written with
respect to frame F0(L) is related to the desired world frame
tip wrench, written with respect to frame F0(0) , by
( )
0
0
( )
( )
[
L
m L
Rp
in which R0Land
frame F0(L) with respect to frame F0(0) . In this case, the
equations must be solved iteratively with respect to both tip
wrench and actuator positions.
0( )F s , must be performed such
. Thirdly, while the external loading
(3)SE
00
( )( )0 n L
== (21)
( )
iL
are used to
(0)
i
are
00
(0)
0
00
000
( )
( )
0
]
F LF
T
L
TT
LL
n Ln L
m L
R
R
(22)
0L
p describe the orientation and position of
III. APPROXIMATE SINGLETUBE MODEL
In contrast to the model presented above, references [11]
[13] propose an approach in which the loaddeflected shape
of a continuum robot is computed as the sequence of two
transformations. The first employs an unloaded kinematic
model to compute the
configuration together with the external loading are the
inputs to a second transformation that computes the
deflected shape by modeling the robot as a single rod with
its stiffness given by the composite stiffness of the robot’s
elements. While approximate since it ignores internal
displacements arising from loading, its solution takes the
form of an initial value problem and so can be computed
robot configuration. This
efficiently.
The equations to be solved are a subset of those for the
multitube model and consist of (2), (17), (18) which are
repeated here for clarity.
( )( )[
( )( )n sf s
00
( )( )u sKs m s
0v s
As before,
m s and
n s are the net bending moment and
shear force on the robot as functions of arc length. Robot
curvature,
u s , described in coordinate frame
algebraically related to
m s . The composite robot
stiffness,
K s , is defined as the effective bending and
torsional stiffness of the robot cross section as a function of
arc length.
The initial robot curvature,
output of the unloaded forward kinematic model. The
boundary conditions for these equations are given by a
subset of (19) consisting of the applied tip force and bending
moment.
( ) body frame external tip moment
( ) body frame external tip forcen L
Since the boundary conditions are all defined at the distal
end of the robot, they can be solved as an initial value
problem by integrating from the tip back to the base.
The solution is, however, subject to the conditions
described in sections II.B.2 and II.B.3 above. Namely, the
equations must be integrated on
tip loading is not defined with respect to the body frame
then the initial value problem must be solved iteratively to
account for the rotation of the body tip frame in response to
deflection. In realtime use, the number of iterations can be
minimized by using the tip frame rotation from the
preceding time step as the initial guess.
00000
0000
( )] [ ( )]
0[
( )
( )( )]
m s
su sv s
u s
m s
n s
(23)
1
00ˆ
u s ( ) ( )
(24)
0( ) 0 1
T
(25)
0( )
0( )
0( )
0( )
F s , is
0( )
0( )
0ˆ ( )
u s , is obtained as the
0
0
m L
(26)
(3) SE
. Furthermore, if the
A. Comparison with Multitube Model
The computational costs of the models can be assessed by
considering the total number of state variables and the
locations of the boundary conditions as summarized in Table
1. While not included in the table, it is also necessary for
both models to simultaneously integrate (
obtain the coordinate frame
F s .
Exclusive of
F s , the total number of state variables is
2n+5 for the multitube model. Since each tube of a robot
contributes two degrees of freedom corresponding to its
rotation and translation, it requires three tubes to produce a
robot with six degrees of freedom. For such a robot, solution
of the multitube model involves integrating eleven state
variables with respect to arc length using split boundary
conditions. The singletube model possesses six states
regardless of the number of tubes and all boundary
conditions are at the distal end.
Since computation of the unloaded kinematic model
)
00
( ),( )v s u s
to
0( )
0( )
2329